Download Computer Science 341 Discrete Mathematics Homework 10

Computer Science 341
Discrete Mathematics
Homework 10
Due at noon on Fri, Dec 6, 2002
No collaboration is permitted for this homework.
Problem 1
Consider a group of nk politicians who must be assigned to n committees. Each politician is qualified
to belong to some but not necessarily all the committees. Each committee must be assigned exactly
k qualified members, and each politician must be assigned to exactly one of the committees for
which that politician qualifies. Give a necessary and sufficient condition on the qualifications of
the politicians for the committees for this to be possible. (Hint: For k = 1 the condition of Hall’s
Theorem suffices.)
Problem 2
Let X be an n-element set and let S1 , S2 , . . . , Sn be subsets of X such that |Si ∩ Sj | ≤ 1 whenever
√
1 ≤ i ≤ j ≤ n. Prove that at least one of the sets Si has size at most C n for some absolute
constant C (independent of n).
Problem 3
The Princeton Tigers are playing a rival team in a 2-out-of-3 series, i.e. they play games until one
team wins a total of two games. The probability that the Tigers win the first game is 1/2. For
subsequent games, the probability of winning depends on the outcome of the preceding game; the
team is energized by victory and demoralized by defeat. If the Tigers win a game, then they have
a 2/3 chance of winning the next game. On the other hand, if the team loses, they have only a 1/3
chance of winning the following game.
(a) What is the probability that the Tigers win the 2-out-of-3 series given that they win the first
game ?
(b) What is the probability that the Tigers win the first game, given that they win the series ?
Problem 4
Let S = {1, 2, . . . , n} and let π be a random permutation on S, i.e. π : S → S is a random bijection
from S to S For a subset A of S, let f (A) be the minimum of π(a) over all elements a ∈ A. Let A
and B be two arbitrary subsets of S. Express the following in terms of A and B.
(a) Pr[f (A) = f (B)]
(a) Pr[f (A) < f (B)]